James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 16.1 Vector Fields 1073
Section 14.6. Notice also that the gradient vectors are long where the level curves are
close to each other and short where the curves are farther apart. That’s because the length
of the gradient vector is the value of the directional derivative of f and closely spaced
level curves indicate a steep graph.
■
A vector field F is called a conservative vector field if it is the gradient of some scalar
function, that is, if there exists a function f such that F − =f . In this situation f is
called a potential function for F.
Not all vector fields are conservative, but such fields do arise frequently in physics.
For example, the gravitational field F in Example 4 is conservative because if we define
then
=f sx, y, zd − −f
−x i 1 −f
−y j 1 −f
−z k
−
2mMGx
sx 2 1 y 2 1 z 2 d 3y2 i 1
− Fsx, y, zd
mMG
f sx, y, zd −
sx 2 1 y 2 1 z 2
2mMGy
sx 2 1 y 2 1 z 2 d 3y2 j 1
2mMGz
sx 2 1 y 2 1 z 2 d 3y2 k
In Sections 16.3 and 16.5 we will learn how to tell whether or not a given vector field is
conservative.
1–10 Sketch the vector field F by drawing a diagram like
Fig ure 5 or Figure 9.
1. Fsx, yd − 0.3 i 2 0.4 j 2. Fsx, yd − 1 2 x i 1 y j
3. Fsx, yd − 2 1 2 i 1 sy 2 xd j
4. Fsx, yd − y i 1 sx 1 yd j
5. Fsx, yd −
y i 1 x j
sx 2 1 y 2
6. Fsx, yd − y i 2 x j
sx 2 1 y 2
7. Fsx, y, zd − i
13. Fsx, yd − ky, y 1 2l
14. Fsx, yd − kcossx 1 yd, xl
I
3
_3 3
_3
II
3
_3 3
_3
8. Fsx, y, zd − z i
III
3
IV
3
9. Fsx, y, zd − 2y i
10. Fsx, y, zd − i 1 k
11–14 Match the vector fields F with the plots labeled I–IV.
Give reasons for your choices.
11. Fsx, yd − kx, 2yl
12. Fsx, yd − ky, x 2 yl
_3 3
_3
_3 3
_3
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